Such a mop press is known from German utility model DE 20 2006 015 209 U1. The mop press serves to squeeze liquids out of a cleaning device, for example, a flat pad or a wet mop. The wringing device comprises a lever that is operatively connected to pressing plates via a shaft. Depending on the configuration, these pressing plates can move in different directions and can also be guided by a connecting link so as to follow a curved path. The pressing plates move against the cleaning device and press the textile part of the cleaning device against a counter-surface, for example, a perforated wringing basket. Such mop presses are usually used commercially. The mop press is affixed either to a mop bucket or to a cleaning trolley. For this purpose, the mop press has a rear fastening mechanism that can be, for example, a hook-shaped projection that forms a U-shaped recess that engages with the bucket or the cleaning trolley in order to affix the mop press. Here, the force that can be applied via the lever onto the wringing device can be limited by the fastening mechanism of the mop press and its counter-piece, for example, by the back wall of the bucket. With conventional designs of the fastening mechanism, the forces and torques generated during the pressing are absorbed in that a first, horizontally positioned counter-piece, for example, an upper tube or the upper edge of the bucket, engages with the U-shaped recess on the press, while a second counter-piece, for example, a lower tube or the surface of the rear wall of the bucket, supports the press vertically below the rear wall of the bucket at a typical distance of approximately 100 mm. If an excessive force if applied via the lever, the mop press can become detached from the fastening mechanism.